What sort of game problems can neural networks trained/evolved with evolutionary algorithms solve, and how are they typically implemented?
Without going in too much detail on how exactly Neural Networks and Generic Algorithms work, I can tell you that both the algorithms are not good candidates for computer games.  They work well in scientific environments where the system is "trained" on a huge data set to adjust the "weights" (variables) for a given problem.  This "training" process requires a lot of processing power, time and a large data set.

Computer games, however either needs to run in real-time (no time for training) or turn-based (not enough data for training).

Another problem is that computer games need to free up as much as possible system resources for physics, graphics, sounds and the user interface to improve the player's experience so game developers usually use other lighter techniques (like a rule-based system) to create the illusion of an AI player.